1. Dynamic Programming Carmine Casciato MUMT 611 Thursday March 31st 2005

2. Overview • Problem Space • Origins • Dynamic Time Warping • Overview of Usage in MIR

3. Problem Space • Multi-stage decision processes • system S characterized as evolution of vector p • N stages * M decisions/stage • multi-dimensional maximization problems

4. Origins • Dynamic Programming (DP) “…the optimal decision to be made at any state of the system. ”Bellman (1957) • “Dynamic” refers to temporal nature of S • Each decision is determined by max/min cost of previous state • Allocation problem, x = y + x-y • fN(x) = Max/Min [g(y) + h(x-y) + fN-1(ay + b(x-y)) 0 <= y <= x

5. Rabiner and Huang 1993 • Dynamic Time Warping (DTW) as solution for time-alignment and normalization of two utterances • (Dis)similarity measurement of two vectors of short-time spectral features is equal to “best” path through feature grid

6. DTW • Path Constraints • endpoint • monotonicity • local path constraints • global path constraints • slope weighting • locally and globally • Dissimilarity metric, constraints, weightings, are all heuristically determined

7. Paulus and Klapuri 2002 • Adopts Rabiner and Huang (1993) DTW to rhythmic similarity • Depends on correct segmentation of rhythms from audio signal • Finds optimal path between feature vectors of loudness and spectral centroid

8. Paulus and Klapuri 2002

9. Usage in MIR • Query by humming • Heo et al. 2003 • Adams et al. 2004 • Nishimura et. al 2001 • Tempo tracking • Raphael 2002 • Feature selection • Chang 1972

10. References • Adams, N., M. Bartsch, J. Shifrin, and G.Wakefield. 2004. Time series alignment for music information retrieval. In Proceedings of the International Conference on Music Information Retrieval: 30310 • Bellman, R. 1957. Dynamic Programming. Princeton: Princeton University Press. • Chang, C. 1972. Dynamic programming as applied to feature subset selection in a pattern recognition system. In Proceedings of the ACM annual conference 1: 94103. • Guo, A., and H. Siegelman. 2004. Time-warped longest common subsequence algorithm for music retrieval. In Proceedings of the International Conference on Music Information Retrieval: 25861. • Heo, S., M. Suzuki, A. Ito, and S. Makino. 2003. Three dimensional continuous DP algorithm for multiple pitch candidates in music information retrieval system. In Proceedings of the International Conference on Music Information Retrieval. • Nishimura, T., H. Hashiguchi, J. Takita, J. Zhang, M. Goto, and R. Oka. 2001. Music signal spotting retrieval by a humming query using start frame feature dependent continuous dynamic programming. In Proceedings of the International Conference on Music Information Retrieval. • Paulus, J., and A. Klapuri. 2002. Measuring the similarity of rhythmic patterns. In Proceedings of the International Conference on Music Information Retrieval. • Raphael, C. 2002. A hybrid graphical model for rhythmic parsing. Artificial Intelligence 137: 217–38.